Edexcel ALevel Maths, Pure Mathematics Paper 2, October 2020 Question Walkthroughs

Pure Paper 2, October 2020, Q1The table below shows corresponding values of x and y for y = square root x / (1 + x). The values of y are given to 4 significant figures.
x 0.5 1 1.5 2 2.5 y 0.5774 0.7071 0.7746 0.8165 0.8452 (a) Use the trapezium rule, with all the values of y in the table, to find an 
estimate for the integral between 0.5 and 2.5 of square root x / (1 + x) dx giving your answer to 3 significant figures.
(b) Using your answer to part (a), deduce an estimate for the integral between 0.5 and 2.5 of square root 9x / (1 + x) dx
Given that the integral between 0.5 and 2.5 of square root 9x / (1 + x) dx = 4.535 to 4 significant figures
(c) comment on the accuracy of your answer to part (b).
(b) Using your answer to part (a), deduce an estimate for the integral between 0.5 and 2.5 of square root 9x / (1 + x) dx
Given that the integral between 0.5 and 2.5 of square root 9x / (1 + x) dx = 4.535 to 4 significant figures
(c) comment on the accuracy of your answer to part (b).

Pure Paper 2, October 2020, Q2Relative to a fixed origin, points P, Q and R have position vectors p, q and r respectively. Given that
● P, Q and R lie on a straight line ● Q lies one third of the way from P to R show that q = 1/3 (r + 2p). 

Pure Paper 2, October 2020, Q3(a) Given that 2 log (4 − x) = log (x + 8) show that x^2 − 9x + 8 = 0
(b) (i) Write down the roots of the equation x^2 − 9x + 8 = 0 (ii) State which of the roots in (b)(i) is not a solution of 2 log (4 − x) = log (x + 8) giving a reason for your answer. 

Pure Paper 2, October 2020, Q4In the binomial expansion of (a + 2x)^7 where a is a constant, the coefficient of x^4 is 15 120.
Find the value of a. 

Pure Paper 2, October 2020, Q5The curve with equation y = 3 × 2^x meets the curve with equation y = 15 − 2^(x+1) at the point P.
Find, using algebra, the exact x coordinate of P. 

Pure Paper 2, October 2020, Q6Given that (x^2 + 8x  3)/(x + 2) = Ax + B + C/(x + 2) find the values of the constants A, B and C
(b) Hence, using algebraic integration, find the exact value of the integral of (x^2 + 8x  3)/(x + 2) between 0 and 6 giving your answer in the form a + b ln 2 where a and b are integers to be found. 

Pure Paper 2, October 2020, Q7Figure 1 shows a sketch of the curve C with equation y = (4x^2 + x) / 2 root x  4ln x
(a) Show that dy / dx = (12x^2 + x  16 root x) / 4x root x The point P, shown in Figure 1, is the minimum turning point on C. (b) Show that the x coordinate of P is a solution of x = (4/3  root x /12)^2/3 (c) Use the iteration formula x1 = 2 to find (i) the value of x2 to 5 decimal places, (ii) the x coordinate of P to 5 decimal places. 

Pure Paper 2, October 2020, Q8A curve C has equation y = f (x)
Given that ● f ʹ(x) = 6x^2 + ax − 23 where a is a constant ● the y intercept of C is −12 ● (x + 4) is a factor of f (x) find, in simplest form, f (x) 

Pure Paper 2, October 2020, Q9A quantity of ethanol was heated until it reached boiling point. The temperature of the ethanol, θ °C, at time t seconds after heating began, is modelled by the equation θ = A − Be^− 0.07t where A and B are positive constants.
Given that 
● the initial temperature of the ethanol was 18 °C
● after 10 seconds the temperature of the ethanol was 44 °C
(a) find a complete equation for the model, giving the values of A and B to 3 significant figures.
Ethanol has a boiling point of approximately 78 °C
(b) Use this information to evaluate the model.
● after 10 seconds the temperature of the ethanol was 44 °C
(a) find a complete equation for the model, giving the values of A and B to 3 significant figures.
Ethanol has a boiling point of approximately 78 °C
(b) Use this information to evaluate the model.

Pure Paper 2, October 2020, Q10(a) Show that cos 3A ≡ 4 cos 3A − 3 cos A
(b) Hence solve, for −90° < x < 180°, the equation 1 − cos 3x = sin 2x 

Pure Paper 2, October 2020, Q11Figure 2 shows a sketch of the graph with equation y = 2  x + 4  − 5
The vertex of the graph is at the point P, shown in Figure 2. (a) Find the coordinates of P. (b) Solve the equation 3x + 40 = 2  x + 4  − 5 A line l has equation y = ax, where a is a constant. Given that l intersects y = 2  x + 4  − 5 at least once, (c) find the range of possible values of a, writing your answer in set notation. 

Pure Paper 2, October 2020, Q12The curve shown in Figure 3 has parametric equations x = 6 sin t, y = 5 sin 2t The region R, shown shaded in Figure 3, is bounded by the curve and the xaxis. (a) (i) Show that the area of R is given by the integral of 60 sin t cos^2 t dt between 0 and pi/2
(ii) Hence show, by algebraic integration, that the area of R is exactly 20 
Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4.
Using the model and given that
● x and y are in metres
● the vertical wall of the dam is 4.2 metres high
● there is a horizontal walkway of width MN along the top of the dam
b) calculate the width of the walkway.
Using the model and given that
● x and y are in metres
● the vertical wall of the dam is 4.2 metres high
● there is a horizontal walkway of width MN along the top of the dam
b) calculate the width of the walkway.

Pure Paper 2, October 2020, Q13The function g is defined by g (x) = 3 ln(x)  7 / ln(x)  2 where x is larger than 0 and x does not equal k where k is a constant.
(a) Deduce the value of k. (b) Prove that gʹ (x) is larger than 0 for all values of x in the domain of g. (c) Find the range of values of a for which g (a) is larger than 0 

Pure Paper 2, October 2020, Q14A circle C with radius r
● lies only in the 1st quadrant ● touches the xaxis and touches the yaxis The line l has equation 2x + y = 12 (a) Show that the x coordinates of the points of intersection of l with C satisfy 5x^2 + (2r − 48) x + (r^2 − 24r + 144) = 0 Given also that l is a tangent to C, (b) find the two possible values of r, giving your answers as fully simplified surds. 

Pure Paper 2, October 2020, Q15A geometric series has common ratio r and first term
a. Given r ≠ 1 and a ≠ 0 (a) prove that Sn =a (1  r^n) / (1  r) Given also that S10 is four times S5 b. find the exact value of r. 

Pure Paper 2, October 2020, Q16Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3.
